Creases caused by internal faces
This issue is caused by having extra edges or faces between contiguous edges or faces in a mesh. In this case, I had extra faces where I extruded off of quad faces.



An example of extra edges is shown, where the orange line is an extra edge because it doesn't contribute to the face data. Edges that don't cross faces and don't contribute to face data will cause the subsurface modifier to have issues, since it checks edges and faces without regard to their usefulness.



(Answered originally by Denis)